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The flow of viscous incompressible fluid over a periodically corrugatedsurface is investigated numerically by solving the Navier-Stokes equation withthe local slip and no-slip boundary conditions. We consider the effective sliplength which is defined with respect to the level of the mean height of thesurface roughness. With increasing corrugation amplitude the effective no-slipboundary plane is shifted towards the bulk of the fluid, which implies anegative effective slip length. The analysis of the wall shear stress indicatesthat a flow circulation is developed in the grooves of the rough surfaceprovided that the local boundary condition is no-slip. By applying a local slipboundary condition, the center of the vortex is displaced towards the bottomthe grooves and the effective slip length increases. When the intrinsic sliplength is larger than the corrugation amplitude, the flow streamlines near thesurface are deformed to follow the boundary curvature, the vortex vanishes, andthe effective slip length saturates to a constant value. Inertial effectspromote vortex flow formation in the grooves and reduce the effective sliplength.

The effective slip length and vortex formation in laminar flow over a rough surface